Then CSG with a positive constant stepsize \(\tau _n=\tau <\tfrac{2}{L}\)  converges to a stationary point of \(J\) .

We want to sketch the proof of theo:ConstantStepConvergence, before going into details. In the deterministic case, lem:descent lemma and lem: char property are used to show that \({J(u_{n+1})\le J(u_n)}\)  for all \(n\in \mathbb {N}\) . It then follows from a telescopic sum argument that \(\Vert u_{n+1}-u_n\Vert \rightarrow 0\) , i.e., every accumulation point of \((u_n)_{n\in \mathbb {N}}\)  is stationary (compare [1]} or [2]}).
